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Let an integer  be bad if there is no way to insert a colon among the digits of  to form a time of day (on a twelve-hour, digital clock), and good if a colon can be inserted to form a time. For example, 100 is good because we can write 1:00, but 164 is bad. How many bad integers between 100 and 200 are equivalent to 3 mod 4?

 Apr 27, 2018
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Hi supermanaccz!

 

So from 100 to 200, we first need to determine what numbers are bad. 

 

From 100 to 159, they are all good, since we can have 1:00 and 1:59.

 

But from 160 to 199, they are all bad. There is no 1:60, since that will be 2 o'clock. 

 

There is no 1: 99.

 

So we need to calculate how many numbers from 160 - 199 are equal to 3 mod 4.

 

The smallest number that is from 160 - 199 that is equal to 3 mod 4 is 163.

 

3 / 4 = 0 R 3

 

163 / 4 = 40 R 3

 

The largest is 199. 

 

199 / 4 = 49 R 3

 

To calculate how many numbers we want from 163 to 199, we do:

 

\((199-163)/3+1=\boxed{13}\)

 

That will be your final answer .

 

I hope this helped, 

 

gavin.

 Apr 27, 2018

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